In this paper, according to the development of the fractional differentiation and its applications in the modern signal processing, we improve the numerical calculation of fractional differentiation by Newton interpolation equation, and propose a new mask, the Newton Interpolation’s Fractional Differentiation (NIFD). Then, we apply this new mask to image edge detection and can obtain the better...

Abstract: Identification of continuous-time non-linear systems characterised by fractional order dynamics is studied. The Riemann-Liouville definition of fractional differentiation is used. A new identification method is proposed through the extension of Hammerstein-type models by allowing their linear part to belong to the class of fractional models. Fractional models are compact and so are us...

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as points and derived differentiation matrices for derivatives. With matrices, were transformed into linear systems, which are easier solve. Two types of numerical simulations, results demo...

This study proposed an image enhancing method which is based on the non-local fractional order differential operator. In this method, a matrix form representation of discrete fractional order differentiation is introduced to enhance the digital image, which is effective to reduce the computation error caused by the traditional local approximate method of the fractional order differentiation. Th...

This paper proposes a generic FPGA realization of an IP core for fractional-order integration and differentiation based on the Grünwald–Letnikov approximation. All dependent terms are approximated to simpler relations using curve fitting enable efficient hardware realization. Compared previous works, proposed design introduces enhancements in range covering both differentiation. An error analys...

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri ...

We prove Cauchy's formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional and differentiation arbitrary nonempty closed sets.

We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional ...